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$\newcommand{\BRD}{\Psi_0} \newcommand{\X}{{\sf X}} \newcommand{\isoto}{\overset\sim\longrightarrow} $Not an answer, but too long for a comment. Let $G$ be a (connected) reductive group over an algebr …

Let $G$ be a group and $M$ be a $G$-module, that is, an abelian group written additively on which $G$ acts: $$ (g,m)\mapsto g m.$$ We consider the group of coinvariants $$ M_G:=G/\langle g m -m\ |\ g\ …

This seemingly elementary question was asked in Mathematics StackExchange.com: https://math.stackexchange.com/q/4779592/37763. It got upvotes, but no answers or comments, and so I ask it here. Let $G$ …

Let $X={\rm Gr}_{n,k,{\Bbb R}}$ denote the Grassmannian of $k$-dimensional subspaces in ${\Bbb R}^n$. We regard $X$ as an ${\Bbb R}$-variety with the set of complex points $X({\Bbb C})={\rm Gr}_{n,k,{ …

$\newcommand{\Z}{{\Bbb Z}} \newcommand{\Q}{{\Bbb Q}} \newcommand{\Hom}{{\rm Hom}} $ The following lemma is certainly known. Lemma (well-known). Let $B$ be a lattice (that is, a finitely generated fre …

$\newcommand{\Hom}{{\rm Hom}} \newcommand{\Gm}{{{\mathbb G}_{m,{\Bbb C}}}} \newcommand{\X}{{\sf X}} $ I am looking for a reference for the following lemma (for which I know a proof): Lemma. Let $\var …

$ \newcommand{\G}{\Gamma} \newcommand{\rsa}{\rightsquigarrow} \newcommand{\Z}{{\mathbb Z}} \newcommand{\Q}{{\mathbb Q}} \newcommand{\Lam}{\Lambda} \newcommand{\Tor}{{\rm Tor}} \newcommand{\Gt}{{\Gamma …

$\newcommand{\Tors}{{\rm Tors}} \newcommand{\tf}{{\rm\, t.f.}} \newcommand{\Gt}{{\Gamma\!,\,\Tors}} \newcommand{\Gtf}{{\Gamma\!,\tf}} \newcommand{\Q}{{\mathbb Q}} \newcommand{\Z}{{\mathbb Z}} \newcomm …

I give an elementary proof of the fact that a quasi-isomorphism of short complexes (complexes of length 2) of $\Gamma$-modules induces an isomorphism on hypercohomology. Actually, it is very close to …

I am looking for a reference for the assertion in the title. In more detail, let $\Gamma=\{1,\gamma\}$ be a group of order 2. Let $A$ be a $\Gamma$-module (an abelian group on which $\Gamma$ acts). T …

Let $X$ be an algebraic variety over an algebraically closed field $k$ of characteristic 0 (a reduced separated scheme of finite type over $k$). Let $G$ be a connected linear algebraic group over $k$ …

I am asking for a reference for the following lemma (for which I know a proof). Lemma. Let $f\colon X\to Y$ be a surjective morphism of irreducible smooth complex algebraic varieties (separated, red …

Let ${\frak g}$ be a simple Lie algebra over $\Bbb C$, and let $\theta$ be an inner involution of ${\frak g}$, that is, an inner automorphism of ${\frak g}$ of order dividing 2. Such automorphisms are …

I am looking for a reference where the relative root system, the relative system of simple roots, and parabolic $\Bbb R$-subgroups for the real algebraic group ${\rm SU}(p,q)$ are explicitly computed. …

I answer Question 1. It is just a calculation. Instead of a real torus, say ${\bf T}$, I consider a pair $(T,\sigma)$, where $T$ is a complex torus and $\sigma\colon T\to T$ is an anti-holomorphic inv …